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Practical Non-linear Energy Harvesting Model and Resource Allocation for SWIPT Systems
Elena Boshkovska, Derrick Wing Kwan Ng, Nikola Zlatanov, Robert Schober
TL;DR
The paper addresses resource allocation for SWIPT when practical energy-harvesting circuits are non-linear rather than linear. It proposes a parametric logistic model and an iterative optimization algorithm, with simulations showing lower harvested power for allocation designed under the conventional linear model.
Problem
Conventional linear energy-harvesting models may mismatch practical circuits whose efficiency varies with input power and whose harvested power saturates.
Method
The paper proposes a parametric logistic non-linear harvesting model and solves the resulting non-convex resource-allocation problem with an iterative algorithm and SDP relaxation.
Results
The proposed model closely matches experimental results, while linear-model-based resource allocation achieves strictly smaller total harvested power because of allocation mismatch.
Takeaways & Limitations
Resource allocation designed for conventional linear harvesting may lead to mismatches for practical non-linear harvesting circuits.
Takeaways & Limitations
The analysis assumes statistically independent channels between energy-harvesting receivers and information receivers.
Abstract
from arXiv · showhide
In this letter, we propose a practical non-linear energy harvesting model and design a resource allocation algorithm for simultaneous wireless information and power transfer (SWIPT) systems. The algorithm design is formulated as a non-convex optimization problem for the maximization of the total harvested power at energy harvesting receivers subject to minimum required signal-to-interference-plus-noise ratios (SINRs) at multiple information receivers. We transform the considered non-convex objective function from sum-of-ratios form into an equivalent objective function in subtractive form, which enables the derivation of an efficient iterative resource allocation algorithm. In each iteration, a rank-constrained semidefinite program (SDP) is solved optimally by SDP relaxation. Numerical results unveil a substantial performance gain that can be achieved if the resource allocation design is based on the proposed non-linear energy harvesting model instead of the traditional linear model.
I. INTRODUCTION
The paper motivates SWIPT as a practical wireless energy-harvesting technology and addresses resource-allocation design issues with a non-linear harvesting model and iterative optimization algorithm.
- RF wireless power transfer enables comparatively controllable energy harvesting for low-power devices and supports simultaneous wireless information and power transfer.
- Prior SWIPT studies addressed rate-energy trade-offs, energy efficiency, power allocation, user scheduling, and subcarrier allocation.
- The proposed approach uses a practical parametric non-linear harvesting model verified with measurement data.
- Resource allocation is formulated as non-convex maximization of total harvested energy and solved optimally by an iterative algorithm.
- Simulation results illustrate harvested-energy loss when resource allocation is designed using a conventional linear harvesting model.
A. Channel Model
The channel model considers a frequency-flat slow-fading downlink multiuser SWIPT system with a multi-antenna transmitter serving single-antenna information receivers and multi-antenna energy receivers.
- The transmitter has NT > 1 antennas and serves K information receivers and J energy harvesting receivers.
- Information receivers are low-complexity single-antenna devices, while each energy harvesting receiver has NR receive antennas.
- In each time slot, the transmitter sends a vector of data symbols to the information receivers.
- The model distinguishes data symbols, beamforming vectors, transmitter-to-information-receiver channels, and transmitter-to-energy-receiver channel matrices.
- Additive white Gaussian noises are modeled at both information and energy receivers, with σ2_s denoting receiver noise power.
B. Energy Harvesting Model
The paper replaces the conventional linear harvesting model with a logistic non-linear model that captures power-dependent efficiency, saturation, and practical circuit effects, then uses it for resource allocation.
- The conventional model assumes harvested energy is directly proportional to received RF power through fixed efficiency ηj.
- Practical harvesting circuits exhibit increasing efficiency at low power, diminishing returns, and limits on maximum harvested energy.
- The proposed model uses a logistic sigmoidal function to capture harvesting dynamics across different input power levels.
- The model includes Ωj for zero-input/zero-output behavior and Mj for the maximum harvested power at saturation.
- Parameters aj, bj, and Mj can be obtained by curve fitting and represent hardware-specific circuit characteristics.
- The proposed model closely matches experimental results, whereas the conventional linear model is inaccurate for non-linear harvesting circuits.
III. PROBLEM FORMULATION AND SOLUTION
The system maximizes total harvested power subject to transmit-power and minimum-SINR constraints, then transforms the non-convex sum-of-ratios objective into an equivalent subtractive form for iterative optimization.
- The design maximizes total harvested power while satisfying the transmitter's maximum-power constraint and each information receiver's minimum required SINR.
- The objective is non-convex because it has a sum-of-ratios form, which prevents direct use of methods designed for single-ratio objectives.
- The sum-of-ratios objective is transformed into an equivalent subtractive objective that yields the same optimal resource allocation policy.
- The resulting algorithm uses nested iterations: an inner optimization for fixed parameters and an outer search for parameters satisfying the associated equations.
A. Solution of the Inner Loop Problem
The inner-loop problem remains non-convex because of a rank-one matrix constraint, so the method removes that constraint to obtain a convex SDP whose relaxation is tight under stated channel conditions.
- The transformed inner-loop problem introduces new and auxiliary optimization variables but remains non-convex because of its rank-one matrix constraint.
- SDP relaxation removes the rank constraint, converting the inner-loop formulation into a convex semidefinite program that can be solved by standard methods.
- With statistically independent channels and a feasible problem, the relaxed SDP has a rank-one optimal beamforming matrix with probability one.
- Under those channel conditions, the SDP relaxation is tight and beamforming is optimal for maximizing harvested power under the proposed non-linear model.
B. Solution of the Outer Loop Problem
The outer-loop algorithm updates the auxiliary variables iteratively using a damped Newton method to satisfy the defining system of equations. The section also notes a one-ER equivalence between linear and non-linear EH policies and the possible extension to dedicated energy beams.
- B. Solution of the Outer Loop Problem: The unique optimal (µ*, β*) is characterized by ϕ(µ, β) = 0 and obtained using a damped Newton update.The method converges to the unique solution satisfying the relevant system of equations.
- B. Solution of the Outer Loop Problem: At iteration n, µ and β are updated using a step size ζ_n and the Newton direction q_n derived from the Jacobian of ϕ(µ, β).ζ_n is selected through a damped step-size rule.
- B. Solution of the Outer Loop Problem: With one ER, the traditional linear and proposed non-linear EH models yield the same optimal resource allocation policy.This is a specific single-ER case identified by the authors.
- B. Solution of the Outer Loop Problem: The signal model can be extended to include dedicated energy beams for ERs by following a similar approach.The extension is stated as a possible modeling direction rather than part of the reported implementation.
IV. RESULTS
The simulations evaluate harvested power under varying SINR requirements, transmit-antenna counts, and numbers of ERs. The proposed non-linear-model design outperforms the linear-model baseline, with the advantage growing as more ERs are present.
- IV. RESULTS: The simulations use K = 2 information receivers, J ERs, 915 MHz carrier frequency, 200 kHz bandwidth, and two receive antennas per ER.The ERs are placed 10 meters from the transmitter and modeled with Rician fading of factor 3 dB.
- IV. RESULTS: Average total harvested power decreases monotonically as the minimum required SINR Γ_req increases.Higher SINR requirements steer transmission toward information receivers, leaving less RF energy for EH.
- IV. RESULTS: Average total harvested power increases with the number of transmit antennas N_T.Additional transmit antennas provide more degrees of freedom for power-efficient resource allocation.
- IV. RESULTS: The proposed non-linear-model design achieves strictly more total harvested power than the conventional linear-model baseline.The baseline can saturate some ERs and underutilize others because it ignores non-linear EH behavior.
- IV. RESULTS: Average total harvested power increases with both the number of ERs J and the number of transmit antennas N_T.The simulations use P_max = 30 dBm and a minimum required SINR of 30 dB for this comparison.
- IV. RESULTS: The proposed scheme’s performance gain over the baseline increases as the number of ERs grows.The reported explanation is that the baseline’s resource-allocation mismatch becomes more pronounced with more ERs.
V. CONCLUSIONS
The paper concludes that practical non-linear EH modeling is important for SWIPT resource allocation, while its optimization is handled through SDP relaxation and structural analysis. The appendix establishes the relevant duality and rank properties used in that solution.
- V. CONCLUSIONS: The proposed EH model captures the non-linear characteristics of practical EH circuits in SWIPT systems.This is the paper’s principal modeling conclusion.
- V. CONCLUSIONS: Resource allocation is formulated as a non-convex sum-of-ratios optimization and solved optimally by an iterative algorithm.The appendix analyzes the SDP-relaxed problem using strong duality, KKT conditions, and dual variables.
- V. CONCLUSIONS: The conclusions identify resource-allocation mismatches when conventional linear EH models are used for practical non-linear EH circuits.The mismatch is the reported consequence of designing allocation algorithms with the linear model.
- V. CONCLUSIONS: The SDP-relaxed formulation supports structural conclusions about the beamforming matrices through complementary slackness and rank analysis.The proof examines null-space structure and conditions under which the optimal beamforming matrix is rank one.
- V. CONCLUSIONS: The appendix’s contradiction argument establishes that B*_k is negative definite with probability one under the stated channel assumptions.The argument uses statistical independence and the finiteness of the primal optimum for finite P_max.